M. Fernanda P. Costa

Practical implementation of an interior point nonmonotone line search filter method

Here we present a primal-dual interior point nonmonotone line search filter method for nonlinear programming. The filter relies on three measures, the feasibility, the centrality and the optimality presented in the optimality conditions, and considers relaxed acceptability criteria for the step size and includes a feasibility restoration phase. Evaluation of the method has, until now, been made on small problems and a comparison is provided with a merit function approach.

An artificial fish swarm algorithm based hyperbolic augmented Lagrangian method

This paper aims to present a hyperbolic augmented Lagrangian (HAL) framework with guaranteed convergence to an @e-global minimizer of a constrained nonlinear optimization problem. The bound constrained subproblems that emerge at each iteration k of the framework are solved by an improved artificial fish swarm algorithm. Convergence to an @e^k-global minimizer of the HAL function is guaranteed with probability one, where @e^k->@e as k->~. Preliminary numerical experiments show that the proposed paradigm compares favorably with other penalty-type methods.

An artificial fish swarm filter-based method for constrained global optimization

An artificial fish swarm algorithm based on a filter methodology for trial solutions acceptance is analyzed for general constrained global optimization problems. The new method uses the filter set concept to accept, at each iteration, a population of trial solutions whenever they improve constraint violation or objective function, relative to the current solutions. The preliminary numerical experiments with a well-known benchmark set of engineering design problems show the effectiveness of the proposed method.

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.

Heuristic-Based Firefly Algorithm for Bound Constrained Nonlinear Binary Optimization

Firefly algorithm (FA) is a metaheuristic for global optimization. In this paper, we address the practical testing of a heuristic-based FA (HBFA) for computing optima of discrete nonlinear optimization problems, where the discrete variables are of binary type. An important issue in FA is the formulation of attractiveness of each firefly which in turn affects its movement in the search space. Dynamic updating schemes are proposed for two parameters, one from the attractiveness term and the other from the randomization term. Three simple heuristics capable of transforming real continuous variables into binary ones are analyzed. A new sigmoid “erf” function is proposed. In the context of FA, three different implementations to incorporate the heuristics for binary variables into the algorithm are proposed. Based on a set of benchmark problems, a comparison is carried out with other binary dealing metaheuristics. The results demonstrate that the proposed HBFA is efficient and outperforms binary versions of differential evolution (DE) and particle swarm optimization (PSO). The HBFA also compares very favorably with angle modulated version of DE and PSO. It is shown that the variant of HBFA based on the sigmoid “erf” function with “movements in continuous space” is the best, in terms of both computational requirements and accuracy.

Assessing the potential of interior point barrier filter line search methods: nonmonotone versus monotone approach

In this article, we present a numerical study of three nonmonotone filter line search techniques, as well as a three-dimensional filter approach, when incorporated into the solver IPOPT, a primal-dual barrier method developed by Wächter and Biegler [On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106 (2006), pp. 25–57] for nonlinear programming. Primary assessment of the proposals has been done with sets of small- and medium-scale problems and large-scale problems separately. Results show that the use of nonmonotone globalization strategies improves efficiency.

An evolutionary algorithm based pattern search approach for constrained optimization

Constrained optimization is one of the popular research areas since constraints are usually present in most real world optimization problems. The purpose of this work is to develop a gradient free constrained global optimization methodology to solve this type of problems. In the methodology proposed, the single objective constrained optimization problem is solved using a Multi-Objective Evolutionary Algorithm (MOEA) by considering two objectives simultaneously, the original objective function and a measure of constraint violation. The MOEA incorporates a penalty function where the penalty parameter is estimated adaptively. The use of penalty function method will enable to further improve the current best solution by decreasing the level of constraint violation, which is made using a gradient free local search method. The performance of the proposed methodology was assessed on a set of benchmark test problems. The results obtained allowed to conclude that the present approach is competitive when compared with other methods available.

Interior point filter method for semi-infinite programming problems

Semi-infinite programming (SIP) problems can be efficiently solved by reduction-type methods. Here, we present a new reduction method for SIP, where the multi-local optimization is carried out with a stretched simulated annealing algorithm, the reduced (finite) problem is approximately solved by a Newtons primal–dual interior point method that uses a novel two-dimensional filter line search strategy to guarantee the convergence to a KKT point that is a minimizer, and the global convergence of the overall reduction method is promoted through the implementation of a classical two-dimensional filter line search. Numerical experiments with a set of well-known problems are shown.

Multilocal programming: a derivative-free filter multistart algorithm

Multilocal programming aims to locate all the local solutions of an optimization problem. A stochastic method based on a multistart strategy and a derivative-free filter local search for solving general constrained optimization problems is presented. The filter methodology is integrated into a coordinate search paradigm in order to generate a set of trial approximations that might be acceptable if they improve the constraint violation or the objective function value relative to the current one. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.

Multistart Hooke and Jeeves filter method for mixed variable optimization

In this study, we propose an extended version of the Hooke and Jeeves algorithm that uses a simple heuristic to handle integer and/or binary variables and a filter set methodology to handle constraints. This proposal is integrated into a multistart method as a local solver and it is repeatedly called in order to compute different optimal solutions. Then, the best of all stored optimal solutions is selected as the global optimum. The performance of the new method is tested on benchmark problems. Its effectiveness is emphasized by a comparison with other well-known stochastic solvers.